Question

In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x - x² and y = x² - x.

Answer 1

See Fig. 

y = 4x - x² = 4 - (x - 2)² 

Therefore, (x - 2)² = -(y - 4) is a parabola with vertex at (2, 4) and upwards.

Again y = x² - x = [x - 1/2]² - 1/4

Therefore, (x - 1/2)² = y + 1/4 is a parabola with vertex B at (1/2, - 1/4) and downwards.

The two parabolas intersect at O(0, 0) and A(5/2, 15/4).

Therefore, ratio = 121/24.1/6 = 121/4.